# Prove that finite dimentional subspaces of normed vector spaces are topologically complemented.

So I'm trying to solve the following exercise:

If $X$ is a normed space and $Y$ a finite dimensional subspace. Show that $Y$ is complemented in $X.$

I know there's a proof of it using the Hahn-Banach Theorem, however I'm trying to prove it without that. My approach is to use the fact that there exists $Z$ such that $Y\bigoplus Z = X$, thus use the projection $P$ onto $Y$, there is a theorem that says that $Y$ is topologically complemented if and only if $P$ is bounded. I thought it would be easy to prove that $P$ is bounded but I can't actually prove it.

• The projection is bounded if and only if $Z$ is closed. But if $X$ is infinite-dimensional, then a nontrivial finite-dimensional subspace always has non-closed algebraic complements. So you cannot prove that the projection is bounded without further assumptions. – Daniel Fischer Oct 21 '17 at 16:02
• So the proof works completely fine if we can find some closed $Z$ that is a complement of $Y$ @DanielFischer – H_Hassan Oct 21 '17 at 16:08
• Well, yes. But finding a closed complement is more or less the task. – Daniel Fischer Oct 21 '17 at 16:10

Theorem: Let $X$ be a NLS, $M$ a subspace and $\phi \in M^*$. Then there exists $\psi \in X^*$ extending $\phi$.